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Quadratic Quest!

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Lesson Plan

Quadratic Quest!

Students will be able to identify quadratic equations, understand their standard form, and begin to explore methods for solving them, connecting these concepts to real-world scenarios.

Understanding quadratic equations is crucial for advanced math and science, providing tools to model projectile motion, optimize designs, and solve complex problems in various fields.

Audience

12th Grade

Time

30 minutes

Approach

Interactive lecture, guided practice, and real-world examples.

Materials

Whiteboard or Projector, Markers or Pens, Quadratic Equations Slides, Quadratic Warm-Up, Quadratic Practice Worksheet, Quadratic Worksheet Answer Key, and Quadratic Cool Down

Prep

Teacher Preparation

15 minutes

Step 1

Warm-Up: Think About It!

5 minutes

  • Distribute the Quadratic Warm-Up to students.
    - Ask students to quickly jot down what they already know about equations and problem-solving.
    - Briefly discuss their initial thoughts as a class.

Step 2

Introduction to Quadratics

10 minutes

  • Present the Quadratic Equations Slides (Slides 1-4).
    - Introduce quadratic equations, their standard form ($ax^2 + bx + c = 0$), and key characteristics.
    - Discuss the importance of the 'a' coefficient.
    - Engage students with questions like: "Where do you think you might see curved paths in real life?"

Step 3

Solving Quadratics: An Overview

10 minutes

  • Continue with Quadratic Equations Slides (Slides 5-7).
    - Briefly introduce different methods for solving quadratic equations (factoring, quadratic formula, completing the square), focusing on factoring simple expressions for this introductory lesson.
    - Guide students through one or two simple examples on the board.
    - Distribute the Quadratic Practice Worksheet and have students work on the first few problems. Provide support as needed.

Step 4

Wrap-Up & Cool Down

5 minutes

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Slide Deck

Quadratic Quest: Unveiling the Parabola!

What are we exploring today?

  • The mysterious world of quadratic equations.
  • Their standard form and what makes them unique.
  • How they help us understand the world around us!

Welcome students and set the stage for a journey into quadratic equations. Ask them to think about curves in real life.

The Standard Form Revealed

Every quadratic equation has a secret identity:

$ax^2 + bx + c = 0$

  • a, b, c: Real numbers
  • a ≠ 0: Why is 'a' so important?
  • x: The variable

Introduce the standard form and explain each variable. Emphasize that 'a' cannot be zero.

Where Do We See Quadratics?

From Sports to Space!

  • Projectile Motion: The path of a thrown ball or rocket.
  • Architecture: The shape of bridges and arches.
  • Engineering: Designing satellite dishes or car headlights.
  • Everyday Life: The arc of water from a fountain.

Discuss real-world examples to make the concept relatable. Encourage students to share their own ideas.

The Beautiful Parabola

The graph of a quadratic equation is a parabola.

  • A symmetrical U-shaped curve.
  • Can open upwards or downwards.
  • Reflects the power of the $x^2$ term.

Briefly mention that graphs of quadratics create parabolas. No deep dive into graphing yet, just an introduction.

Cracking the Code: Solving Quadratics

How do we find 'x'?

  • Factoring: Breaking down the expression.
  • Quadratic Formula: A universal tool! ($x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$)
  • Completing the Square: A methodical approach.

Today, we'll dip our toes into factoring!

Transition to solving methods. Mention the three main ways, but focus on factoring for now as an entry point.

Let's Factor! (Example 1)

Example: Solve $x^2 - 5x + 6 = 0$

  • Find two numbers that multiply to 6 and add to -5.
  • These numbers are -2 and -3.
  • $(x-2)(x-3) = 0$
  • Therefore, $x=2$ or $x=3$.

Walk through a simple factoring example. Encourage student participation.

Your Turn! (Example 2)

Try to solve: $x^2 + 7x + 10 = 0$

  • What two numbers multiply to 10 and add to 7?




(Answer: $x=-5$ or $x=-2$)

Provide another example for students to try or work through together.

Recap: What Did We Discover?

Today, we:

  • Identified quadratic equations and their standard form.
  • Explored real-world applications.
  • Briefly touched on solving by factoring.

Next time:

  • Diving deeper into solving methods!
  • More exciting quadratic adventures!

Summarize the key takeaways and preview what's next. Reinforce the real-world connections.

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Warm Up

Quadratic Quest: Warm-Up Challenge!

Instructions:

Take 3-5 minutes to answer the following questions to the best of your ability. Don't worry if you don't know all the answers - this is just to get our brains warmed up!

1. What comes to mind when you hear the word "equation"?







2. Can you think of any real-life situations where something follows a curved path? Describe one.







3. If you had an equation like $2x + 4 = 10$, how would you go about finding the value of 'x'?







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Worksheet

Quadratic Quest: Practice Worksheet

Instructions:

Work through the problems below. Show your work for each problem. Remember, the goal is to find the values of 'x' that make the equation true!

Part 1: Identifying Standard Form

For each equation, identify the values of $a$, $b$, and $c$ in the standard form $ax^2 + bx + c = 0$.

  1. $x^2 + 3x + 2 = 0$
    $a = \underline{\hspace{2cm}}$
    $b = \underline{\hspace{2cm}}$
    $c = \underline{\hspace{2cm}}$



  2. $2x^2 - 5x + 1 = 0$
    $a = \underline{\hspace{2cm}}$
    $b = \underline{\hspace{2cm}}$
    $c = \underline{\hspace{2cm}}$



  3. $x^2 - 9 = 0$
    $a = \underline{\hspace{2cm}}$
    $b = \underline{\hspace{2cm}}$
    $c = \underline{\hspace{2cm}}$



  4. $3x^2 + 6x = 0$
    $a = \underline{\hspace{2cm}}$
    $b = \underline{\hspace{2cm}}$
    $c = \underline{\hspace{2cm}}$



Part 2: Solving by Factoring (Introduction)

Solve the following quadratic equations by factoring. Show all your steps.

  1. $x^2 + 5x + 6 = 0$











  2. $x^2 - 7x + 12 = 0$











  3. $x^2 + 2x - 8 = 0$











  4. $x^2 - x - 20 = 0$











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Answer Key

Quadratic Quest: Practice Worksheet Answer Key

Part 1: Identifying Standard Form

For each equation, identify the values of $a$, $b$, and $c$ in the standard form $ax^2 + bx + c = 0$.

  1. $x^2 + 3x + 2 = 0$

    • Thought Process: Compare the given equation to $ax^2 + bx + c = 0$. The coefficient of $x^2$ is $a$, the coefficient of $x$ is $b$, and the constant term is $c$.
    • $a = 1$
    • $b = 3$
    • $c = 2$



  2. $2x^2 - 5x + 1 = 0$

    • Thought Process: Similar to the above, identify the coefficients for $x^2$, $x$, and the constant term.
    • $a = 2$
    • $b = -5$
    • $c = 1$



  3. $x^2 - 9 = 0$

    • Thought Process: Rewrite the equation as $x^2 + 0x - 9 = 0$ to clearly see the coefficients.
    • $a = 1$
    • $b = 0$
    • $c = -9$



  4. $3x^2 + 6x = 0$

    • Thought Process: Rewrite the equation as $3x^2 + 6x + 0 = 0$ to clearly see the coefficients.
    • $a = 3$
    • $b = 6$
    • $c = 0$



Part 2: Solving by Factoring (Introduction)

Solve the following quadratic equations by factoring. Show all your steps.

  1. $x^2 + 5x + 6 = 0$

    • Thought Process: We need two numbers that multiply to 6 and add to 5. These numbers are 2 and 3.
    • $(x + 2)(x + 3) = 0$
    • Set each factor equal to zero and solve:
      • $x + 2 = 0 \implies x = -2$
      • $x + 3 = 0 \implies x = -3$
    • Solutions: $x = -2$, $x = -3$











  2. $x^2 - 7x + 12 = 0$

    • Thought Process: We need two numbers that multiply to 12 and add to -7. These numbers are -3 and -4.
    • $(x - 3)(x - 4) = 0$
    • Set each factor equal to zero and solve:
      • $x - 3 = 0 \implies x = 3$
      • $x - 4 = 0 \implies x = 4$
    • Solutions: $x = 3$, $x = 4$











  3. $x^2 + 2x - 8 = 0$

    • Thought Process: We need two numbers that multiply to -8 and add to 2. These numbers are 4 and -2.
    • $(x + 4)(x - 2) = 0$
    • Set each factor equal to zero and solve:
      • $x + 4 = 0 \implies x = -4$
      • $x - 2 = 0 \implies x = 2$
    • Solutions: $x = -4$, $x = 2$











  4. $x^2 - x - 20 = 0$

    • Thought Process: We need two numbers that multiply to -20 and add to -1. These numbers are -5 and 4.
    • $(x - 5)(x + 4) = 0$
    • Set each factor equal to zero and solve:
      • $x - 5 = 0 \implies x = 5$
      • $x + 4 = 0 \implies x = -4$
    • Solutions: $x = 5$, $x = -4$











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Cool Down

Quadratic Quest: Cool Down Ticket

Instructions:

Answer the following questions to help me understand what stuck with you today!

1. In your own words, what is a quadratic equation?







2. Why is the 'a' term in $ax^2 + bx + c = 0$ important (i.e., what happens if $a=0$)?







3. Name one real-world example where quadratic equations might be used.







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